A Hydrodynamical Geometrization of Matter and Chronometricity in General Relativity - Indranu Suhendra

A Hydrodynamical Geometrizationof Matter and Chronometricity

in General Relativity

Indranu Suhendro

Abstract: In this work, we outline a new complementary modelof the relativistic theory of an inhomogeneous, anisotropic universewhich was rst very extensively proposed by Abraham Zelmanov toencompass all possible scenarios of cosmic evolution within the frame-work of the classical General Relativity, especially through the devel-opment of the mathematical theory of chronometric invariants. Indoing so, we propose a fundamental model of matter as an intrinsic

exural geometric segment of the cosmos itself, i.e., matter is modelledas an Eulerian hypersurface of world-points that moves, deforms, andspins along with the entire Universe. The discrete nature of matter isreadily encompassed by its representation as a kind of discontinuitycurvature with respect to the background space-time. In addition, ourpresent theoretical framework provides a very natural scheme for theunication of physical elds. An immediate scale-independent partic-ularization of our preliminary depiction of the physical plenum is alsoconsidered in the form of an absolute monad model corresponding toa universe possessing absolute angular momentum.

Contents:

x1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102x2. The proposed geometrization of matter: a cosmic monad . . . . 105x3. Reduction to the pure monad model . . . . . . . . . . . . . . . . . . . . . . . . 111x4. Hydrodynamical unication of physical elds . . . . . . . . . . . . . . . .114x5. A Machian monad model of the Universe. . . . . . . . . . . . . . . . . . . .116x6. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

Dedicated to Abraham Zelmanov

x1. Introduction. The relativistic theory of a fully inhomogeneous,anisotropic universe in the classical framework of Einstein's GeneralTheory of Relativity has been developed to a fairly unprecedented,over-arching extent by the general relativist and cosmologist AbrahamZelmanov [1]. The ingenious methodology of Zelmanov has been pro-foundly utilized and developed in several interesting physical situations,shedding further light on the intrinsic and extensive nature of the clas-

STKIP Surya, Surya Research and Education Center, Scientia Garden, GadingSerpong, Tangerang 15810, Indonesia. E-mail: [email protected]

Indranu Suhendro 103

sical General Relativity as a whole (see [2]).By the phrase \classical General Relativity", we wish to emphasize

that only space-time and gravitational elds have been genuinely, cohe-sively geometrized by the traditional Einsteinian theory. Nevertheless,the construction of the mathematical theory of chronometric invariantsby Zelmanov enables one to treat General Relativity pretty much in thecontext of some kind of four-dimensional continuum mechanics of thevery substance (plenum) of space-time geometry itself.

We factually note, in passing, that several independent theoreticalapproaches to the geometric unication of space-time, matter, and phys-ical elds, in both the extensively classical sense and the non-classicalsense, have been constructed by the Author elsewhere (for instance,see [3] and the bibliographical list of the Author's preceding works |diverse as they are | therein).

In the present work, we are singularly concerned with the method-ology originally outlined by Zelmanov. Nevertheless, fully acquaintedwith the powerful depth, elegance, and beauty of his work, we shallstill present some newly emerging ideas by rst-principle construction,as well as some well-established understandings afresh, while uniquelysituating ourselves in the alleyway wherefrom both the cosmos and theclassical General Relativity are insightfully envisioned by Zelmanov.

As such, we shall theoretically ll a few gaps in the fabric of theclassical General Relativity in general, and of Zelmanov's methodol-ogy in particular, by proposing a fundamental hydrodynamical modelof matter, so as to possibly substantiate the material structure of theobserver in common with the preferred, stable cosmic reference framewith respect to which the observer is at rest, i.e., one that co-moves,co-deforms, and co-rotates with respect to the entire Universe.

Indeed, we shall proceed rst by geometrizing matter and discoveringa natural way to reectively superimpose the small-scale picture uponthe entire Universe, yielding a unied description of the observer andthe cosmos.

In the very general sense (far from the usual homogeneous, isotropiccosmological situations), we may note at this point that not all observerscan automatically be qualied as fundamental observers, i.e. \observa-tional monads", with respect to whose observation the structure of theUniverse intrinsically appears the way it is observed by them.

Such, of course, is true also for observers assuming a homogeneous,isotropic universe and observing it accordingly. However, in certaincases incorporating, e.g., the absolute rotation of the universe, the prob-lem of true interiority (and structural totality) arises in the sense that

104 The Abraham Zelmanov Journal | Vol. 3, 2010

we can no longer recognize certain innate properties of the universein the reference frames specic to homogeneous, isotropic models only.Such frames may be slowly translating and deforming to keep them-selves at the natural expansive rate of the universe, but in the presenceof self-rotation (intrinsic angular momentum), a physical system is quitesomething else to be accounted for in itself. For, if certain elementaryparticles (as we know them) are truly elementary, we shall know thetotal sense of observation of the Universe also from within their (com-mon) interior and ultimately discover that, irrespective of scales, theUniverse is self-contained in their very existence.

Now, recalling that which lies at the heart of the theory of chrono-metric invariants, we may posit further that the interior (and the totalpossible exterior) of the Universe can only be known by a rather ad-vanced non-holonomic observer, i.e., one who is not merely \incidental"to the mesoscopic scale of (seemingly homogeneous) ordinary things, butone who builds his system of reference with respect to the interior andexterior of things in the required extreme limits, i.e., by rather direct in-depth cognition of the logically self-possible meta-Universe, beyond anyself-limited experimental set-up. In other words, the totality of the lawsof cognition is intrinsic to such an observer endowed with a \syntacticaltotality of logical operators" (a whole contingency of self-reexive log-ical grammar). This, in turn, necessarily belongs to the interior of thedirectly observable (perceptual) Universe. One can then see how thissubstantially diers from a mere \bootstrap" universe.

Hence, regarding observation, our \anthropic principle with furtherself-qualication" is true only for observers dynamically situating them-selves in certain unique non-holonomic frames of reference bearing thespecic characteristics of motion of very elementary microscopic objects(such as certain elementary particles) and macroscopic objects exhibit-ing natural chronometricity with respect to the whole Universe (suchas certain spinning stars, planets, galaxies, and metagalaxies). This,then, would be true for individual observers as well as an aggregate ofcommon observers | such as those situated on a special rotating (plan-etary) islet of mass | in their own unique (\universally preferred")non-holonomic coordinate systems.

Such observers are truly situated at the world-points of the respec-tive Eulerian hypersurfaces (representing matter) in common with theentire non-holonomic, inhomogeneous, anisotropic Universe. This is be-cause, while \inhering" in matter itself, they automatically possess allthe geometric material congurations intrinsic to both matter itself andthe entire Universe.


In our theory, as we shall see, the projective chronometric struc-ture plays the role of the geometrized non-Abelian gauge eld strength.Hence, any natural extension of the study will center around the corre-sponding emphasis that the inner constitution of elementary particlescannot be divorced from chronometricity (the way it is hydrodynam-ically geometrized here). This, like in the original case of Zelmanov,gives one the penetrating condence to speak of elementary particles,in addition to large-scale objects, purely in the framework of the chrono-metric General Relativity, as if without having to mind the disparitiesinvolving scales (of particles and galaxies).

Here, chronometricity is geometrized in such a way that co-substan-tial motion results, both hydrodynamically and geometrically, from thefundamental properties of the extrinsic curvature of the material hyper-surface (i.e., matter itself).

In addition, the Yang-Mills curvature is generalized by the presenceof the asymmetric extrinsic curvature, as in [5]. Only in pervasive at-ness does it go into the usual Yang-Mills form of the Standard Model(whose background space-time is Minkowskian). We shall not employthe full form of the particular Finslerian connection as introduced in [3],but only the respective metric-compatible part, with the correspondinggeodesic equation of motion intrinsically generating the generally co-variant Lorentz equation.

Hence, while encompassing the elasticity of space-time, we shall fur-ther advance the notion of a discontinuous Eulerian hypersurface suchthat it geometrically represents matter and chronometricity at once, andsuch that it may be applied to any cosmological situation independentlyof scales.

Indeed, as we shall see, the Machian construction (see x5 herein)is a special condition for \emergent inertia", without having to invokeboth Newtonian absolute (external) empty space and a distant refer-ence frame. Rather, the whole process is meant to be topologicallyscale-independent. An alternative objective of the present approach,therefore, is such that the structure of General Relativity, when de-veloped (generalized) this way, can apparently meet that of quantumtheory in a parallel fashion.

x2. The proposed geometrization of matter: a cosmic monad.Let us consider an arbitrary orientation of a mobile, spinning hypersur-face C3(t)= @4 as the boundary of the world-tube 4 of geodesics inthe background Universe M4. Denoting the regular boundary by B3(t)and the discontinuity hypersurface cutting through 4 by , we see


that C3(t)=B3(t)[. We emphasize that C3(t) is a natural geometricsegment of M4, i.e., it is created purely by the dynamics of the intrinsic(and global) curvature and torsion of M4. This is to the extent that theunit normal vector with respect to C3(t) is immediately given by theworld-velocity u(s) along the world-line s.

We call C3(t) a monad, i.e., a substantive Eulerian structure of mat-ter. As we shall see, this dynamical monad model is fully intrinsic to thefabric of space-time, i.e., inseparable from (not external to) the intrinsicstructure of the Universe, thereby allowing us to incorporate the sub-sequent geometrization of matter (and material elds) into Einstein'seld equation.

Our substantial depiction of matter lling the cosmos also implies thewave-like nature of the hypersurface C3(t), for the velocity eld of thepoints of C3(t) | representing individual group particles | is no longersingly oriented. This allows us to project the fundamental materialstructure pervasively outward | onto the Universe itself. Consequently,this model readily applies to all sorts of observers, other than just a co-moving one (whose likeness we shall especially refer to as the \purelymonad observer").

As we know, the innitesimal world-line, along which C3(t) moves,is explicitly given by the metric tensor g(x) of M

4 as

ds2 = g dxdx = g00 dx

0dx0 + 2g0idx0dxi + gik dx

idxk =

= c2d2 d2; (2.1)where we denote the speed of light as c. The proper time, the generallynon-holonomic, evolutive spatial segment (the hypersurface segment),the metric tensor of the hypersurface, and the linear velocity of spacerotation (i.e., of material spin) are respectively given by

d =g0dx

cpg00

=pg00 dt+

g0icpg00

dxi =pg00 dt 1

c2vidx

i; (2.2)

d =phik dxidxk ; (2.3)

hik = gik + g0ig0kg00

= gik + 1c2

vivk ; (2.4)

vi = c g0ipg00

: (2.5)

Einstein's summation convention is utilized with space-time Greek indices run-ning from 0 to 3 and projective material-spatial Latin indices from 1 to 3.


Denoting the unit normal vector of the material hypersurface by N,we see that N=u= dx

ds , and especially that

Ni = ui = 1cvi : (2.6)

In a simplied matrix representation, we therefore have

g =

g00 Ni

pg00

Nipg00 hik +NiNk

!: (2.7)

Now, the fundamental projective relation between the backgroundspace-time metric g(x) and the global material metric hik(x; u) isreadily given as

g = h + uu ; (2.8)where, with f (vi; dt)! vif (dt) and fvi; @@t! vif @@t,

h =@Y i

@x@Y k

@x(gik) ; (2.9)

dY i = dxi + f (vi; dt) ; (2.10)

@

@Y i=@x

@Y i@

@x=

@

@xi+ f

vi;

@

@t

; (2.11)

h = + uu ; h h = uu ; (2.12)

hi = h

@Y i

@x= @Y

i

@x; (2.13)

hi hi =

uu ; hihk = ik ; (2.14)

h u = 0 ; hiu

= 0 : (2.15)

Let us represent the natural basis vector of M4 by g and that ofC3(t) by !i. We immediately obtain the generally asymmetric extrinsiccurvature of C3(t) through the inner product

Zik =

u;

@!i@Y k

(2.16)

i.e.,Zik = urkhi = hi hk r u ; (2.17)

where r denotes covariant dierentiation, i.e., for an arbitrary tensoreldQab:::cd:::(x) and metric-compatible connection form

amk(x), presented


herein with arbitrary indexing,

rkQab:::cd::: =@Qab:::cd:::@xk

+ amkQmb:::cd::: +

bmkQ

am:::cd::: + : : :

mckQab:::md::: mdkQab:::cm::: : : : ; (2.18)

abc =1

2gam

@gmb@xc

@gbc@xm

+@gcm@xb

+ a[bc]

gamgbn

n[mc] + gcn

n[mb]

; (2.19)

DQab:::cd:::ds

= uereQab:::cd::: : (2.20)

Henceforth, round and square brackets on indices shall indicate sym-metrization and anti-symmetrization, respectively.

Hence, we see that the extrinsic curvature tensor of the materialhypersurface is uniquely expressed in terms of the four-dimensional ve-locity gradient tensor given by the expression

' = r u ; (2.21)i.e.,

Zik = hi hk ' : (2.22)This way, we have indeed geometrized the tensor of the rate of ma-

terial deformation and the tensor of material vorticity ! , as canbe seen from the respective symmetric and anti-symmetric expressionsbelow:

Z(ik) = hi hk ; Z[ik] = hi hk ! ; (2.23)where

=1

2

r u+ru ; ! = 12

r uru : (2.24)Meanwhile, noting immediately that

rkhi = Ziku; (2.25)we obtain the following relation:

@hi@Y k

= pikhp hi hk Ziku: (2.26)

Both pik (Yp) of C3(t) and (x) of M

4 are generally asymmetric,non-holonomic connection forms. We see that they are related to each


other through the following fundamental relations:

pik = hp

@hi@Y k

hp hi hk ; (2.27)

=hi

@hi@x

hppikhihk+Zikhihku+u@u@x

Zikhi hku : (2.28)

The associated curvature tensor of C3(t), Ri jkl(

ijl), and that of M

4,R (

), are then respectively given by

Ri jkl =@ijl@Y k

@

ijk

@Y l+pjl

ipk pjkipl ; (2.29)

R =@@x

@

@x+

; (2.30)

where, as usual,rlrk rkrlQab:::cd::: == RmcklQ

ab:::md::: +R

mdklQ

ab:::cm::: + Ra mklQmb:::cd:::

Rb mklQam:::cd::: 2m[kl]rmQab:::cd::: ; (2.31)rbra rarb = 2c[ab]rc ; (2.32)where is an arbitrary scalar eld.

At this point, we obtain the complete projective relations betweenthe background space-time geometry and the geometric material space.The relations are as follows:

Rijkl = ZikZjl ZilZjk + hi hj hkhl R + Sjklhi ; (2.33)

rlZik rkZil = uhi hkhl R 2p[kl]Zip + uSikl : (2.34)

In terms of the curvature tensor Ri jkl(

ijl) of C

3(t), and that of M4,which is R (

), with the segmental torsional curvature (incorpo-

rating possible analytical discontinuities as well) given by

Sijk =@

@Y j

@hi@Y k

@@Y k

@hi@Y j

+ h

i

@hj@Y k

@h

k

@Y j

: (2.35)

Now, we can four-dimensionally express the (generalized generallycovariant) gravitational force F, the spatial deformation D , and theangular momentum A in terms of our geometrized material deforma-


tion and material vorticity as follows:

F = 2c2u! ; (2.36)

D = chh

; (2.37)

A = chh

! ; (2.38)

such that

= chh

' = D +A ; (2.39)

Dik = hi h

k() ; (2.40)

Aik = hi h

k[] : (2.41)

As a result, we obtain the geometrized dynamical relation

Rijkl = hi h

j h

kh

l

R+''''

+ hi Sjkl : (2.42)

Furthermore, let us introduce the Eulerian (substantive) curvatureof the material hypersurface which satises all the natural symmetriesof the curvature and torsion tensors of the background space-time asfollows:

Fijkl = hi h

j h

kh

l R + h

i Sjkl : (2.43)

We immediately see that

Fijkl = Rijkl 1c

DikDjl DilDjk +AikAjl AilAjk +

+DikAjl DilAjk +AikDjl AilDjk; (2.44)

and, in addition, we also obtain the inverse projective relations

R Ruu Ruu RuuRuu Ruuuu Ruuuu Ruuuu Ruuuu == hjh

kh

l

hi(RijklZikZjl+ZilZjk)+SjkluuSjkl

; (2.45)

Ru Ruuu Ruuu =

= hihkhlrlZik rkZil + 2p[kl]Zip Siklu : (2.46)

The complete geometrization of matter in this hydrodynamical ap-proach represents a continuum mechanical description of space-time


where the extrinsic curvature of any material hypersurface manifestsitself as the gradient of its velocity eld. As such, the geometric eldequations simply consist in specifying the world-velocity of the movingmatter (especially directly from reading o the components of the fun-damental metric tensor). The acquisition of individual particles, as aspecial case of the more general group particles, is immediately at handwhen the material hypersurface enclosing a volumetric segment of thecosmos is small enough, i.e., in this case the particles are ordinary in-nitesimal space-time points translating and spinning in common withthe deforming and spinning Universe on the largest scale.

x3. Reduction to the pure monad model. Having formulatedthe general structure of our scheme for the substantive geometrizationof matter (as well as physical elds, essentially by way of our precedingworks as listed in [3]) in the preceding section, we can now explicitlyarrive at the cosmological picture of Zelmanov for general relativisticdynamics, i.e., the theory of chronometric invariants.

Much in parallel with Yershov [4], we may simply state the strongmonad model of the cosmos of Zelmanov as follows:

1) The Universe as a whole spins, inducing the spin of every elemen-tary constituent in it;

2) The Universe is intrinsically inhomogeneous, anisotropic, and non-holonomic, giving rise to its diverse elementary constituents (i.e.,particles) on the microscopic scale, including its specic funda-mental properties (e.g., mass, charge, and spin);

3) The small-scale structure of the Universe is simply holographic(\isomorphic") to the large-scale cosmological structure, therebyrendering the Universe truly self-contained;

4) The linear velocity (or momentum) of any microscopic or macro-scopic object is essentially induced by the global spin of the Uni-verse, such that the individual motion of matter is none other thanthe segmental motion of the Universe.

We shall refer to the above conventions as the pure monad model.Consequently, we have the chronometrically invariant condition rep-

resented byfvi; dt

= 0 ; (3.1)

f

vi;

@

@t

6= 0 : (3.2)

Therefore, with respect to the material hypersurface C3(t), we see


that

hik = gik + g0ig0kg00

= gik + 1c2vivk ; (3.3)

hik = gik; hi = i ; (3.4)vi = c g0ip

g00; vi = cpg00 g0i; (3.5)

v2 = vivi = hik v

ivk ; (3.6)

u0 =pg00 ; u

0 =1pg00

; (3.7)

ui =g0ipg00

= vic; ui = 0 ; (3.8)

dY i = dxi: (3.9)

In our theory, Zelmanov's usual dierential operators of chronomet-ricity are given by

@

@Y i=

@@xi

=@

@xi+

1

c2vi

@@t

; (3.10)

@@t

=1pg00

@

@t; (3.11)

@

@Y i

@@t

@@t

@

@Y i=

@2

@xi@t

@2

@t@xi=

1

c2Fi

@@t

; (3.12)

@2

@Y k @Y i @

2

@Y i@Y k=

@2

@xk @xi

@2

@xi@xk= 2

c2Aik

@@t

: (3.13)

Here the three-dimensional gravitational-inertial force, material de-formation, and angular momentum are simply given by the three-dimen-sional chronometrically invariant components of the four-dimensionalquantities F, D , and A of the preceding x2 | in the case of van-ishing background torsion | as follows:

Fi =1pg00

@w

@xi @vi

@t

; (3.14)

Dik =1

2

@hik@t

; Dik = 12

@hik

@t; (3.15)

Aik =1

2

@vk@xi

@vi@xk

+

1

2c2Fivk Fkvi

; (3.16)


where the associated Zelmanov identities are

@Aik@Y l

+@Akl@Y i

+@Ali@Y k

=@Aik@xl

+@Akl@xi

+@Ali@xk

=

= 1c2AikFl +AklFi +AliFk

; (3.17)

@Aik@t

=1

2

@Fi@Y k

@Fk@Y i

=

1

2

@Fi@xk

@Fk@xi

; (3.18)

and the gravitational potential scalar is

w = c2 (1pg00) : (3.19)The symmetric chronometrically invariant connection of Zelmanov

can be given here by

ikl=

i(kl)+h

iphkq

q[pl]+hlq

q[pk]

=1

2hip@hpk@Y l

@hkl@Y p

+@hlp@Y k

=

=1

2hip@hpk

@xl

@hkl@xp

+@hlp@xk

=

=1

2hip@hpk@xl

@hkl@xp

+@hlp@xk

+

1

c2DikvlDklvi+Dil vk

: (3.20)

Note that while the extrinsic curvature tensor Zik is naturally asym-metric in our theory (in order to account for geometrized material vor-ticity), we might impose symmetry upon the material connection pikwhenever convenient (or else we can associate its anti-symmetric part,through projection with respect to the background torsion, with theelectromagnetic and chromodynamical gauge elds, as we have done,e.g., in [3] and [5]).

Now, with respect to the geometrized dynamical relations of thepreceding section, we obtain

R = hjh

kh

l

hi(RijklZikZjl+ZilZjk)+SjkluuSjkl

+

+uXuX+uYuY+uuJuuJ++uuKuuK ; (3.21)

where

X =R0pg00

; Y =R0pg00

; (3.22)

J =R00g00

; K =R00g00

= J : (3.23)


These quantities, whose three-dimensional components may be link-ed with Zelmanov's various three-dimensional curvature tensors [1, 2],appear to correspond to certain generalized currents.

Further calculation reveals that

X + uJ uJ == hihkhl

rlZik rkZil + 2p[kl]Zip Sikl u

; (3.24)

Y =2pg00p

g00 1uJ uJ

: (3.25)

Therefore, the immediate general signicance of these currents lies inthe dynamical formation of matter itself with respect to the backgroundstructure of the world-geometry (represented by M4).

x4. Hydrodynamical unication of physical elds. In this sec-tion, we shall deal with the explicit structure of the connection formunderlying the world-manifold M4, as well as that of matter | the ma-terial hypersurface C3(t), by recalling certain fundamental aspects of ourparticular approach to the geometric unication of physical elds out-lined in [3] and [5], which very naturally gives us the correct equation ofmotion for a particle (endowed with structure) moving in gravitationaland electromagnetic elds while internally also experiencing the Yang-Mills gauge eld, i.e., as an intrinsic geodesic equation of motion givenby the following generalized metric-compatible connection form:

= (x; u) =

1

2g@g@x

@g@x

+@g@x

+

+e

2mc2Fu

FuF u+Sg

S+S

: (4.1)

The anti-symmetric electromagnetic eld tensor F is fully geo-metrized through the relation

F = 2mc2

e[]u ; (4.2)

whose interior structure is given by the geometrized Yang-Mills gaugeeld [5], here in terms of the internal material coordinates of C3(t) as

F i = 2hi[] =@Ai@x

@Ai

@x+ i g^ i klA

kA

l + 2Z

ikA

k[u ] ; (4.3)

F =mc2

eF i ui ;

i[kl] =

1

2i g^ i kl ; (4.4)


where Ai=hi is the gauge eld strength (not to be confused withthe angular momentum), g^ is a coupling constant, and i kl is the three-dimensional permutation tensor.

The material spin tensor S is readily identied here through theanti-symmetric part of the four-dimensional form of the extrinsic cur-vature ' (i.e., the material vorticity !) of M

4:

S = S u S u ; (4.5)

S = s^'[] = s^! =1

2s^r uru ; (4.6)

where s^ is a constant spin coecient, which can possibly be linked tothe electric charge e, the mass m, and the speed of light in vacuum c,and hence to the Planck-Dirac constant ~ as well, such that we canexpress the connection form more compactly as

=1

2g@g@x

@g@x

+@g@x

+

+e

2mc2Fu

F u F u+ 2S u : (4.7)

Therefore, owing to the fully intrinsic dynamics of the geometrizedphysical elds located in M4, i.e.,

Du

ds= ur u = 0 ; (4.8)

we see that the following condition is naturally satised:

S u = 0 (4.9)

in addition to the equation of motion

mc2du

ds+ u

u= eF u

; (4.10)

where the usual connection coecients are

=1

2g@g@x

@g@x

+@g@x

: (4.11)

Meanwhile, from x2, we note that' = r u = hihkZik ; (4.12)

Zik = urkhi ; (4.13)' u

= 0 ; ' u = 0 ; (4.14)


and so we (re-)obtain

pik = hp

@hi@Y k

hphi hk ; (4.15)

= hi

@hi@x

hppikhihk 'u + u@u@x

+ ' u ; (4.16)

where the material connection of C3(t) can be explicitly expressed as

ikl =1

2hip@hpk@Y l

@hkl@Y p

+@hlp@Y k

+1

2i g^ i kl ; (4.17)

or, in other words,

ikl =1

2hip@hpk@xl

@hkl@xp

+@hlp@xk

+

+1

c2Dikvl Dklvi +Dil vk

+1

2i g^ i kl : (4.18)

This way, we have also obtained the fundamental structural formscorresponding to the immediate structure of our geometric theory ofchiral elasticity [6], which, to a certain extent, is capable of encompass-ing the elastodynamics of matter in our present theory, as representedby the material hypersurface C3(t).

x5. A Machian monad model of the Universe. We shall nowturn towards developing a particular pure monad model, i.e., one inwhich the Universe possesses absolute angular momentum such thatmatter arises entirely from the intrinsic inhomogeneity and anisotropyemerging from the non-orientability and discontinuity of the very geom-etry of the material hypersurface C3(t) with respect to the backgroundspace-time M4. This goes down to saying that the cosmos has neither\inside" nor \outside" as graphically outlined in [4], and that each pointin space-time indeed possesses intrinsic informational spin, irrespectiveof whether or not its corresponding empirical constitution possesses ex-trinsic angular momentum.

Recall, from the previous section, that the anti-symmetric part ofthe material connection form is given by the complex expression

i[kl] =1

2i g^ i kl ; (5.1)

which displays the internal constitution of matter in terms of the gaugecoupling constant g^. Now, the four-dimensional permutation tensor is


readily given byikl u = hi hkhl ; (5.2)

i.e.,

= iklhihkhlu + a + b + c ; (5.3)ikl = hi hkhl u ; (5.4)a = u

u ; (5.5)

b = uu ; (5.6)

c = uu : (5.7)

We therefore see that

i[kl] = 1

2i g^ h

ih

kh

l u

; (5.8)

and, in particular, that

rmi[kl] = 1

2i g^ hih

kh

l h

m

'

: (5.9)

The spatial curvature giving rise to matter can now be written as

Ri jkl = Bi jkl +M

i jkl ; (5.10)

B i jkl =@P ijl@Y k

@Pijk

@Y l+ Pmjl P

imk PmjkP iml ; (5.11)

M i jkl = r^kCijl r^lCijk + Cmjl Cimk CmjkCiml ; (5.12)

P ikl =1

2hip@hpk@Y l

@hkl@Y p

+@hlp@Y k

= ikl ; (5.13)

Cikl =

i[kl] hip

hkm

m[pl] + hlm

m[pk]

= i[kl] ; (5.14)

where r^ denotes covariant dierentiation with respect to the symmetricconnection form P ikl(

ikl).

The special integrability conditions for our particular model of space-time geometry possessing absolute angular momentum will be given by

hi = 0 ; (5.15)

hi hj h

kh

l

R + '' ''

= 0 ; (5.16)

such that, explicitly,

ikl = hi

@hk@Y l

= hi

@hk@xl

= hi

@hk@xl

+1

c2vl

@hk@t

: (5.17)


We therefore obtain

=1

2u g

@u@x

@u@x

+dgds

+1

2u

g

@u

@x @u

@x

; (5.18)

i.e.,

=1

2u g

dgds

+ u

g[]u + u

[]

(5.19)

such that the world-velocity u plays the role of a fundamental \metricvector".

This way, matter (material curvature), and hence inertia, arises pu-rely from the segmental torsional (discontinuity) curvature as follows:

Ri jkl = hi

@

@Y l

@hj@Y k

@@Y k

@hj@Y l

; (5.20)

i.e.,

Ri jkl = 2

c2hiAkl

@hj@t

; (5.21)

where the angular momentum Aik is given by

Aik =1

2

@vk@xi

@vi@xk

+

1

2c2(Fivk Fkvi)

uc[ik] +

vipg00

[0k] +vkpg00

[i0]

; (5.22)

Fi =1pg00

@w

@xi @vi

@t

+ 2

c2pg00

[0i]u ; (5.23)

vi = c g0ipg00

= c ui ; ui = 0 ; (5.24)

u0 =pg00 ; u

0 =1pg00

; (5.25)

w = c2 (1pg00) : (5.26)In this particular scheme, therefore, every constitutive object in the

Universe spins in the topological sense of gaining informational spinfrom the very formation of matter itself. Inertia would then be a prop-erty of matter directly arising from this intrinsic mechanism of spin,which encompasses the geometric formation of all massive objects atany scale. This, in turn, subtly corresponds to the Machian conjec-ture of the inertia (mass) of an object being dependent on a distant,


massive frame of reference (if not all other massive objects in the Uni-verse). However, since our peculiar geometric mechanism here exists atevery point of space-time, and in the topological background of things,the corresponding generation of inertia is simply more intrinsic thanthe initial Machian scheme. Accordingly, there is no need to invokethe existence of a distant galactic frame of reference, other than thegeneral non-orientability and curvature-generating discreteness of thehypersurface representing matter.

x6. Conclusion. We have outlined a seminal sketch of a fully hydro-dynamical geometric theory of space-time and elds, which might com-plement Zelmanov's chronometric formulation of the General Theory ofRelativity. In our theory, chronometricity is particularly geometrizedthrough the unique hydrodynamical nature of the asymmetric extrinsiccurvature of the material hypersurface.

Following our previous works we have unied the gravitational andelectromagnetic elds, with chromodynamics arising from the fully ge-ometrized inner structure of the electromagnetic eld, which is shown tobe the Yang-Mills gauge eld (appearing here in its generalized form).In the present work, it is interesting to note that the role of the non-Abelian gauge eld (represented by its components, namely, Ai) is verynaturally played by the projective chronometric structure (with compo-nents hi), and so the inner constitution of elementary particles cannotbe divorced from chronometricity at all.

In our approach to Mach's principle through a pure monad modelpossessing absolute angular momentum, the unique Kleinian topologyof the Universe gives rise to inertia in terms of the non-orientable spindynamics and discrete intrinsic geometry of the material hypersurface,rendering the respective generation of inertia both local and global (i.e.,signifying, in a cosmological sense, scale-independence as well as intrin-sic topological interdependence among \particulars" and \universals").

Submitted on November 27, 2010

1. Zelmanov A. L. Chronometric Invariants: On Deformations and the Curva-ture of Accompanying Space. Translated from the preprint of 1944, AmericanResearch Press, Rehoboth (NM), 2006.

2. Borissova L. and Rabounski D. Fields, Vacuum, and the Mirror Universe. 2ndedition, Svenska fysikarkivet, Stockholm, 2009.

3. Suhendro I. A new Finslerian unied eld theory of physical interactions.Progress in Physics, 2009, vol. 4, 81{90.


4. Yershov V.N. Fermions as topological objects. Progress in Physics, 2006,vol. 1, 19{26.

5. Suhendro I. A unied eld theory of gravity, electromagnetism, and the Yang-Mills gauge eld. Progress in Physics, 2008, vol. 1, 31{37.

6. Suhendro I. On a geometric theory of generalized chiral elasticity with discon-tinuities. Progress in Physics, 2008, vol. 1, 58{74.

Vol. 3, 2010 ISSN 1654-9163

THE

ABRAHAM ZELMANOV

JOURNALThe journal for General Relativity,gravitation and cosmology

TIDSKRIFTEN

ABRAHAM ZELMANOVDen tidskrift for allmanna relativitetsteorin,

gravitation och kosmologi

Editor (redaktor): Dmitri RabounskiSecretary (sekreterare): Indranu Suhendro

The Abraham Zelmanov Journal is a non-commercial, academic journal registeredwith the Royal National Library of Sweden. This journal was typeset using LATEXtypesetting system. Powered by Ubuntu Linux.

The Abraham Zelmanov Journal ar en ickekommersiell, akademisk tidskrift registr-erat hos Kungliga biblioteket. Denna tidskrift ar typsatt med typsattningssystemetLATEX. Utford genom Ubuntu Linux.

Copyright c The Abraham Zelmanov Journal, 2010All rights reserved. Electronic copying and printing of this journal for non-prot,academic, or individual use can be made without permission or charge. Any part ofthis journal being cited or used howsoever in other publications must acknowledgethis publication. No part of this journal may be reproduced in any form whatsoever(including storage in any media) for commercial use without the prior permissionof the publisher. Requests for permission to reproduce any part of this journal forcommercial use must be addressed to the publisher.

Eftertryck forbjudet. Elektronisk kopiering och eftertryckning av denna tidskrifti icke-kommersiellt, akademiskt, eller individuellt syfte ar tillaten utan tillstandeller kostnad. Vid citering eller anvandning i annan publikation ska kallan anges.Mangfaldigande av innehallet, inklusive lagring i nagon form, i kommersiellt syfte arforbjudet utan medgivande av utgivarna. Begaran om tillstand att reproducera delav denna tidskrift i kommersiellt syfte ska riktas till utgivarna.

Documents

A Hydrodynamical Geometrization of Matter and Chronometricity in General Relativity - Indranu Suhendra